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21 PRODUCT OF MEASURES 241 Let j?, q be such that \<>p< q< +00. Show that, if U is of weak type (p}p) and also of weak type (q, q\ then U is of type (r, r) (Section 13.17, Problem 7) for all r such that p<r<q (Marcinkiewicz's interpolation theorem). (For each /e £^(X, ju,) and a > 0, let/« be the function which is equal to f(x) if \f(x)\ > a, and zero otherwise, and put/« = //«. Show that, for each / > 0, Integrate from 0 to oo and use Problem 1 .) R*(/) = sup Rk(/),